You mention a British program. I do not know about the specific features you mention, but a British company called Serif used to make very highly-regarded DTP software.
You could generate some sort of hash of the selected options, encode it with something that’s short and human readable and then share that on the screen?
Something easy for a human to transfer like honey-chair-balcony or something. Store the settings in the backend against this identifier and then you can retrieve it later? So you still need a backend but no need for accounts
I think they were bought by Corel
Edit: Serif are in the article, sorry